We obtain large n asymptotics of n×n Hankel determinants whose weight has a one-cut regular potential and Fisher-Hartwig singularities. restrict our attention to the case where associated equilibrium measure possesses either one soft edge hard (Laguerre-type) or two edges (Jacobi-type). also present some applications in theory random matrices. In particular, we can deduce from results for parti...

We give a simple proof of central limit theorem for linear statistics the circular $\beta $-ensembles which is valid at almost microscopic scales functions class $C^{3}$. Using coupling introduced by Valko and Virag [48], we deduce Sine$_{\beta }$ processes. also discuss connections between our result theory Gaussian Multiplicative Chaos. Based on results [37], show that exponential logarithm r...

Theoretical and experimental investigations of chaos synchronization and its application to chaotic data transmissions in semiconductor lasers with optical feedback are presented. Two schemes of chaos synchronization—complete and generalized synchronization—are discussed in the delay differential systems. The conditions for chaos synchronization in the systems and the robustness for the paramet...

We consider a number of possible roles of complex dynamics and chaos in information processing by neural networks. First, we review the working principles of some well-known neural networks, and then discuss a number of approaches to utilization of chaos in neural networks. Our main goal is to present a novel view of the problem of chaos in information processing. We demonstrate that chaos emer...

Let X be a compact metric space, let φ be a continuous self-map on X , and let F(X) denote the space of fuzzy sets on X equipped with the levelwise topology. In this paper we study relations between various dynamical properties of a given (crisp) dynamical system (X,φ) and its Zadeh’s extension Φ on F(X). Among other things we study various (weak, strong, mild etc.) mixing properties and also s...

Lasers are essentially chaotic system. However, semiconductor lasers are classified into stable class B lasers. Meanwhile, they are easily destabilized by external perturbations (introduction of extra degree of freedom). We discuss chaos in lasers and also instability and chaos in semiconductor lasers. Recently, new semiconductor lasers with extra device structures have been developed and they ...

This paper aims at a historical and pedagogical presentation of some important contributions of the research on thermonuclear fusion by magnetic confinement to the study of Hamiltonian chaos. This chaos is defined with the help of Poincaré maps on a simple two-wave Hamiltonian system. A simple criterion for computing the transition to large scale chaos is introduced. A renormalization group app...

The theory of chaos and shadowing developed recently by the author is amplified to the case of a heteroclinically tubular cycle. Specifically, let F be a C3 diffeomorphism on a Banach space. F has a heteroclinically tubular cycle that connects two normally hyperbolic invariant manifolds. Around the heteroclinically tubular cycle, a Bernoulli shift dynamics of submanifolds is established through...

Hyper chaos is the non-periodic behaviour of deterministic nonlinear dynamical system that is highly sensitive to initial conditions with more than one positive Lypunov Exponents, i.e., its dynamics expand not only small line segments, but also small area elements, these giving rise to a ‘thick’ chaotic attractor. Bifurcation effects include the period doubling route to chaos, the intermittency...

This paper studies chaos synchronization in the modified Chua's circuit with x x using linear state feedback control scheme. A new sufficient condition is derived for chaos synchronization between the two modified Chua's circuits with x x based on the Lyapunov stability theory and the Linear Quadratic Optimal Control theory. A simulation example is given for demonstration. 1. Introduction Today...